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Abstract

When leveraging orthogonal polynomials for describing freeform optics, designers typically focus on the computational efficiency of convergence and the optical performance of the resulting designs. However, to physically realize these designs, the freeform surfaces need to be fabricated and tested. An optimization constraint is described that allows on-the-fly calculation and constraint of manufacturability estimates for freeform surfaces, namely peak-to-valley sag departure and maximum gradient normal departure. This constraint’s construction is demonstrated in general for orthogonal polynomials, and in particular for both Zernike polynomials and Forbes 2D-Q polynomials. Lastly, this optimization constraint’s impact during design is shown via two design studies: a redesign of a published unobscured three-mirror telescope in the ball geometry for use in LWIR imaging and a freeform prism combiner for use in AR/VR applications. It is shown that using the optimization penalty with a fixed number of coefficients enables an improvement in manufacturability in exchange for a tradeoff in optical performance. It is further shown that, when the number of coefficients is increased in conjunction with the optimization penalty, manufacturability estimates can be improved without sacrificing optical performance.

Fig. 2 For the three-mirror telescope, optical performance and manufacturability estimates versus square-sum penalty weight with (top) Zernikes and (bottom) 2D-Qs. All values of the lower-bound designs are within 5% of the benchmark.

Fig. 3 Layout for (left) the benchmark design and square-sum penalized designs with 2D-Qs for weights (middle-left) 10−2, (middle-right) 100, and (right) 102. The layout changes from the benchmark to the weight 10−2 and weight 100 designs, but not from the weight 100 design to the weight 102 design. The Zernike designs (not shown) have a similar trend.

Fig. 5 Location of the centered and outermost 3 mm sub-pupils used for evaluating optical performance within the 8 mm eyebox. Performance was also evaluated over sub-pupils located halfway between the outermost and center sub-pupils (not shown).

Fig. 7 For the reflective prism, optical performance versus manufacturability estimates as square-sum penalty changes with (top) Zernikes and (bottom) 2D-Qs. All values of the lower-bound design are within 5% of the benchmark.

Fig. 8 Layout of Zernike prism designs for (left) the unpenalized benchmark and (right) the highest-weight square-sum penalized designs. Notice the change in layout. The 2D-Q designs (not shown) show a similar trend.